IPhone – Where has WhatsApp gone

It sounds very much like the phone wasn't wiped & set up as your own new device.
This means that you will need the old login data for everything you need to do on the phone - this is not good.

You will need to get your cousin to make sure the phone is removed from his iCloud account, then you need to wipe it & start afresh.
You will lose all existing data, but currently, nothing on the phone belongs to you.

See What to do before selling or giving away your iPhone, iPad, or iPod touch

He can only view your info if he's logged into WhatsApp using your ID and password. It doesn't matter under which account the app was downloaded/updated from the App Store.

When you're done with the iPhone, and before you give it back to your friend, be sure to log out of WhatsApp.